Re ning Dynamics of Gene Regulatory Networks in a ... · PDF fileThe analysis of stable states...

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Refining Dynamics of Gene Regulatory Networks in a Stochastic π -Calculus Framework Lo¨ ıc Paulev´ e, Morgan Magnin, Olivier Roux To cite this version: Lo¨ ıc Paulev´ e, Morgan Magnin, Olivier Roux. Refining Dynamics of Gene Regulatory Net- works in a Stochastic π-Calculus Framework. Transactions on Computational Systems Biology, Springer, 2011, XIII, pp.171-191. <10.1007/978-3-642-19748-2 8>. <hal-00397235v2> HAL Id: hal-00397235 https://hal.archives-ouvertes.fr/hal-00397235v2 Submitted on 18 Feb 2014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Transcript of Re ning Dynamics of Gene Regulatory Networks in a ... · PDF fileThe analysis of stable states...

Refining Dynamics of Gene Regulatory Networks in a

Stochastic π-Calculus Framework

Loıc Pauleve, Morgan Magnin, Olivier Roux

To cite this version:

Loıc Pauleve, Morgan Magnin, Olivier Roux. Refining Dynamics of Gene Regulatory Net-works in a Stochastic π-Calculus Framework. Transactions on Computational Systems Biology,Springer, 2011, XIII, pp.171-191. <10.1007/978-3-642-19748-2 8>. <hal-00397235v2>

HAL Id: hal-00397235

https://hal.archives-ouvertes.fr/hal-00397235v2

Submitted on 18 Feb 2014

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Refining Dynamics of Gene Regulatory

Networks in a Stochastic π-Calculus Framework

Loıc Pauleve, Morgan Magnin, Olivier Roux

IRCCyN, UMR CNRS 6597,Ecole Centrale de Nantes, France

{loic.pauleve,morgan.magnin,olivier.roux}@irccyn.ec-nantes.fr

Abstract. In this paper, we introduce a framework allowing to modeland analyse efficiently Gene Regulatory Networks (GRNs) in their tem-poral and stochastic aspects. The analysis of stable states and inferenceof Rene Thomas’ discrete parameters derives from this logical formalism.We offer a compositional approach which comes with a natural transla-tion to the Stochastic π-Calculus. The method we propose consists insuccessive refinements of generalised dynamics of GRNs. We illustratethe merits and scalability of our framework on the control of the differ-entiation in a GRN generalising metazoan segmentation processes, andon the analysis of stable states within a large GRN studied in the scopeof breast cancer researches.

Changes since http://dx.doi.org/10.1007/978-3-642-19748-2_8

– Add a 6= b in V equation of Definition 9 (Hitless Graph).

1 Introduction

Modelling, analysis and numerical or stochastic simulations are a usual meansto predict the behaviour of complex living systems such as interacting genes.

Regulations between genes (activation or inhibition) are generally representedby Gene Regulatory Network (GRN) graphs. However, a GRN graph is notenough to describe dynamics. In continuous frameworks such as ordinary dif-ferential equations, parameters for differential equations are needed. In logical(or qualitative) frameworks such as boolean or discrete networks, dynamics aredriven by Rene Thomas’ parameters or equivalent [1].

Hybrid modelling brings quantitative aspects — such as temporal or stochas-tic parameters — to logical modelling. In the field of formal languages, κ lan-guage [2] or Stochastic π-Calculus [3,4,5] bring theoretical Computer Scienceframeworks for biological modelling. In the field of formal verifications of bio-logical systems, frameworks like Time or Stochastic Petri Nets [6,7], Biocham[8], Timed Automata [9], Charon (Hybrid Modelling) [10] and Linear HybridModelling [11] bring the first bricks for verifying and controlling dynamics ofsuch systems.

2 L. Pauleve, M. Magnin and O. Roux

Inference of temporal and stochastic parameters is still challenging as thedomain of parameters is continuous and its volume generally grows exponen-tially with the number of genes. Compositional approaches, inherent to processalgebras, aspire at reducing this complexity by allowing a local reasoning.

Our aim is temporal parameters synthesis for verifying formal properties onhybrid models.

Our contribution consists in the introduction of both temporal and stochasticparameters into process algebra models of GRNs through a new Stochastic π-Calculus framework: the Process Hitting framework.

Starting from a GRN without any other parameters, its largest dynamicsare expressed in Process Hitting and then are refined to match the expectedbehaviour. Such a refinement is achieved by constructing cooperativity betweengenes and by creating stable states. As we will show, detecting stable states of aProcess Hitting is straightforward, as well as infering the Rene Thomas’ discreteparameters K.

Moreover, the Stochastic π-Calculus naturally brings time and stochasticityinto our Process Hitting framework. We introduce a stochasticity absorption fac-tor to highlight either the temporal or stochastic aspect of reactions. The directtranslation of Process Hitting to the Stochastic π-Calculus allows simulations ofsuch models by softwares like BioSpi [3] or SPiM [12].

Several points motivate the choice of the Stochastic π-Calculus frameworkfor introducing the Process Hitting. The Stochastic π-Calculus can be consid-ered as a “low-level” process algebra: there are very few operators compared,for instance, to the Beta Workbench [13] or Bio-PEPA [14]. This makes thepresentation of the Process Hitting as a Stochastic π-Calculus framework moregeneric. Moreover, the Stochastic π-Calculus comes with a bunch of establishedtools as previously cited and translations into various framework have alreadybeen formalized, such as to PRISM, a probabilistic model checking tool [15,16].

This paper is structured as follows. Section 2 introduces our framework andhow it is used to build the generalised dynamics of a GRN. Section 3 presentsdynamics refinement techniques and Section 4 shows how infering the ReneThomas’ parameters leading to such dynamics. Introduction of both temporaland stochastic parameters within Process Hitting models is addressed in Sec-tion 5. The overall approach is illustrated by two applications in Section 6. Thefirst applies the refinement method to a toy GRN involved in biological segmen-tations phenomena. The temporal and stochastic parameters are then infered tobring a particular behaviour to the system. The second application shows thescalability of the Process Hitting framework by modelling a large GRN composedof 20 genes.

Notations Given a set S,

n︷ ︸︸ ︷

S × · · · × S, will be abbreviated as Sn. If S is finiteand countable, we note |S| its cardinality. Given a n-tuple C, C[x/y] refers tothe n-tuple C within the element y has been substituted by x. Belonging andcartesian product for n-tuples are defined similarly to sets. [xi;xi+n] refers tothe interval {xi, xi+1, . . . , xi+n}. ’∧’ stands for the logical and connector.

Refining Dynamics of Gene Regulatory Networks 3

2 Generalised Dynamics for Gene Regulatory Networks

First, we recall the basis of the Rene Thomas’ discrete modelling frameworkfrom which we designed our refinement approach. This method is described insubsections 2.2 and 2.3. This leads to a straightforward translation into theπ-Calculus which makes it possible to express generalised dynamics of GRNs.

2.1 Gene Regulatory Networks

GRNs are often described by interaction graphs where nodes are genes with acti-vation and inhibition relations respectively represented by positive and negativeedges [1,17].

In the discrete framework of Rene Thomas, each gene has at least two qual-itative levels. The influence of an activating (resp. inhibiting) gene on its targetdepends on a threshold value: when the level of the gene is greater or equal thanthe threshold, the gene holds a positive (resp. negative) effect; when the level ofthe gene is lower than the threshold, the gene holds a negative (resp. positive)effect [1].

Definition 1 (Gene Regulatory Network Graph). A Gene RegulatoryNetwork graph is a triple (Γ,E+, E−) where Γ is the finite set of genes and

at−→ b ∈ E+ (resp. E−), t positive integer, if and only if the gene a above level t

is an activator (resp. inhibitor) for b. We note ai the level i of the gene a.

Given a GRN graph (Γ,E+, E−), the maximum qualitative level for genea ∈ Γ is noted ala where la is the highest threshold involved in its regulations:

la = max({t | ∃b ∈ Γ, at−→ b ∈ E+ ∪ E−}) . (1)

We denote levels+(a, b) (resp. levels−(a, b)) the set of levels of a where aeffectively activates (resp. inhibits) b.

Definition 2 (Effective Levels). If at−→ b ∈ E+, levels+(a, b) = [at; ala ] and

levels−(a, b) = [a0; at−1]. If at−→ b ∈ E−, levels+(a, b) = [a0; at−1] and

levels−(a, b) = [at; ala ]. Else levels+(a, b) = levels−(a, b) = ∅.

2.2 The Process Hitting Framework

We want to describe the action of a gene at a given level on another one. Ifthe gene a at a given level i is an activator for b, it has a positive action on b,meaning the level of b will tend to increase. Conversely if a at a level i′ is aninhibitor for b, it has a negative action on b and then the level of b will tend todecrease.

The action is “a at level i making b at level j increase (or decrease) to levelk”. We say ai hits bj to make it bounce to bk and note this action ai → bj � bk.In the process hitting framework, ai, bj , bk are refered as processes and a, b assorts. Sorts can represent genes, but also logical entities, as described in furthersections.

4 L. Pauleve, M. Magnin and O. Roux

Definition 3 (Action). An action is noted ai → bj � bk where ai is a processof sort a and bj 6= bk two processes of sort b. ai → bj is the hit part, and bj � bkthe bounce part. When ai = bj, such an action is refered as a self-action and aiis called a self-hitting process.

In this paper, one and only one process of each sort is present at any instant.Hence, hits between different processes of a same sort are prohibited. The set ofthese living processes gives the state of the Process Hitting.

Definition 4 (Process Hitting). A Process Hitting PH is a triple (Σ,L,H):

– Σ = {a, b, . . . } is the finite countable set of sorts,– L =

a∈Σ La is the set of states for PH, with La = {a0 . . . ala} the finite andcountable set of processes of sort a ∈ Σ and la a positive integer, a 6= b ⇒ai 6= bj ∀(ai, bj) ∈ La × Lb,

– H = {ai → bj � bk, · · · | (a, b) ∈ Σ2, (ai, bj , bk) ∈ La × Lb × Lb, bj 6= bk, a =b ⇒ ai = bj}, is the finite set of actions.

At a given state s ∈ L, an action ai → bj � bk is playable if both processes aiand bj are present in s. When this action is played, the process bk replaces bj .

Definition 5 (Next States). Let (Σ,L,H) be a Process Hitting and s ∈ L beone of its states. The set of the next possible states for s are computed as follows:

next(s) = {s[bk/bj ] | ∃(ai, bj) ∈ s2, ∃bk ∈ Lb, ai → bj � bk ∈ H} .

Definition 6 (Stable state). Let PH = (Σ,L,H) be a Process Hitting ands ∈ L be a state, s is a stable state for PH if and only if next(s) = ∅.

2.3 Graphical Representations of a Process Hitting

We set up two complementary graphical representations of a Process Hitting.The first one exhibits the actions between process levels, the second one pointsout the absence of hits between them. We finally define the State Graph of aProcess Hitting. Figure 1 shows an instance for each of these three representa-tions.

Given a Process Hitting (Σ,L,H), its Hypergraph represents each actionai → bj � bk ∈ H by a directed hyperedge from ai to bk passing by bj . The hitpart (ai to bj) is drawn as a plain edge and the bounce part (bj to bk) as a dottededge.

Definition 7 (Process Hitting Hypergraph). The Hypergraph of a ProcessHitting (Σ,L,H) is a couple (P,A) where P =

a∈Σ La are the vertices andA ⊆ P 3 the directed hyperedges given by A = {(ai, bj , bk) | ai → bj � bk ∈ H}.

In the following we introduce a complementary representation we call HitlessGraph. It will allow us to obtain extra results such as the stable states of aProcess Hitting (Section 3.2). The Hitless Graph of a Process Hitting (Σ,L,H)

Refining Dynamics of Gene Regulatory Networks 5

relates two processes of different sorts if and only if they hit neither each othernor themselves. Vertices of a Hitless Graph may be split into n ≤ |Σ| partitionshaving no element inside related to each other: a partition is, for any sort, asubset of its processes without self-actions. Such a graph is called n-partite.

Definition 8 (n-Partite Graph). A graph G = (V,E) is n-partite if and onlyif V =

⋃nk=1 Vk, Vk 6= ∅, ∀1 ≤ k, k′ ≤ n, Vk ∩ Vk′ = ∅ and (ai, bj) ∈ E ⇒

∃1 ≤ k 6= k′ ≤ n, ai ∈ Vk ∧ bj ∈ Vk′ .

Definition 9 (Hitless Graph). Given a Process Hitting PH = (Σ,L,H), itsHitless Graph PH = (V,E) is defined as a non-directed graph where the verticesV and edges E are computed as follows:

V =⋃

a∈Σ

{ai ∈ La | ∀ai′ ∈ La ∄ ai → ai � ai′ ∈ H}

E = {(ai, bj) ∈ V 2 | a 6= b ∧∀bj′ ∈ Lb ∄ ai → bj � bj′ ∈ H

∧ ∀ai′ ∈ La ∄ bj → ai � ai′ ∈ H} .

Property 1. By construction of V and E, PH is a n-partite graph, n ≤ |Σ|,where each partition is a subset of processes for one and only one sort and toeach sort corresponds at most one partition.

We also define the n-cliques of a graph which are subsets of n vertices suchthat each element is related to each other.

Definition 10 (n-Clique). Given a graph G = (V,E), C ⊆ V is a |C|-cliqueof G if and only if ∀(ai, bj) ∈ C2, {ai, bj} ∈ E.

Property 2. n-cliques of a n-partite graph have one and only one vertex in eachpartition.

Finally, the State Graph of a Process Hitting represents the possible transi-tions between each couple of its states.

Definition 11 (State Graph). Given a Process Hitting (Σ,L,H), its StateGraph is a directed graph S = (L,E ⊆ L2) with (s, s′) ∈ E ⇔ s′ ∈ next(s).

2.4 From Process Hitting to the π-Calculus

A main advantage of our approach is its natural translation to the π-Calculus.In this subsection we propose a method to translate any Process Hitting into aπ-Calculus expression.

We briefly present the fragment of the π-Calculus which is sufficient for trans-lating a Process Hitting. The full syntax for π-Calculus and examples can befound in [5,18]. π-Calculus expressions compose two kinds of objects: indepen-dently defined processes and channels shared by some processes. A process Phas the capability to output (resp. input) on a channel γ and then become P ′,

6 L. Pauleve, M. Magnin and O. Roux

a0

a1

b0

b1

b2

a0

a1

b0

b1

b2

a0b0 a1b0

a0b1 a1b1

a0b2 a1b2

(a) Hypergraph (b) Hitless Graph (c) State Graph

Fig. 1. Graphical representations for the Process Hitting PH = ({a, b}, {a0, a1} ×{b0, b1, b2},H) with H = {b0 → a0 � a1, a1 → b0 � b1, a1 → b1 � b2}.

noted !γ.P ′ (resp. ?γ.P ′). Output and input are synchronized operations, i.e. anoutputting process is blocked until another process inputs on the same chan-nel. A process can also execute an internal action (τ), nil operation (0) or oneamongst several (P ′ + P ′′).

Let PH = (Σ,L,H) be a Process Hitting. For each process ai of PH, a π-Calculus process Ai is defined as follows. For each action ai → bj � bk ∈ H wherea 6= b, a new channel γα is created. The π-Calculus process Ai has the ability tooutput on this channel and the π-Calculus process Bj has the ability to inputon this channel so as to become Bk (2). For each self-action ai → ai � aj ∈ H,Ai has the ability to become Aj after performing an internal action τα (3).

Ai ::=∑

α=ai → bj � bk∈H

a 6=b

!γα.Ai +∑

α=bj → ai � ak∈H

a 6=b

?γα.Ak (2)

+∑

α=ai → ai � ak∈H

τα.Ak (3)

2.5 Generalised Dynamics for Gene Regulatory Networks

Our method to analyse GRNs takes benefit from the use of refinement techniques.Starting from the largest set of possible dynamics for the GRN, we graduallytake into account only the specified behaviours and exclude the other ones, thusleading to a restrictive process.

We call this largest set of dynamics the generalised dynamics for the GRNgraph. It is described by the following rules: the level of a gene increases (resp.decreases) if and only if at least one of its activators (resp. inhibitors) is present.The absence of activators is equivalent to the presence of one inhibitor.

Let G = (Γ,E+, E−) be a GRN graph. For all (a, b) ∈ Γ 2, we build the setof actions Hb

a from a to b reflecting the rules above:

– If at−→ b ∈ E+, all processes of sort a below the threshold t hit all processes

of sort b but b0 to make them decrease to the level below. Moreover, allprocesses of sort a above the threshold t hit all processes of sort b but blb to

Refining Dynamics of Gene Regulatory Networks 7

make them increase to the level above:

Hba = {ai → bj � bj−1 | 0 ≤ i < t, 1 ≤ j ≤ lb}

∪ {ai′ → bj′ � bj′+1 | t ≤ i′ ≤ la, 0 ≤ j′ < lb} .

– If at−→ b ∈ E−, the actions are defined similarly to the previous case except

for the bounce directions which are reversed:

Hba = {ai → bj � bj+1 | 0 ≤ i < t, 0 ≤ j < lb}

∪ {ai′ → bj′ � bj′−1 | t ≤ i′ ≤ la, 1 ≤ j′ ≤ lb} .

– If b = a and ∄c ∈ Γ, ct−→ b ∈ E− ∪E+, gene b lives in absence of activators:

all processes of sort b but b0 hit themselves to decrease to the level below.

Hbb = {bi → bi � bi−1 | 1 ≤ i ≤ lb} .

– Obviously, if at−→ b /∈ E−∪E+ for any t and the previous case does not hold,

we define Hba = ∅.

The Process Hitting for the generalised dynamics of G is given by

PH = (Γ,∏

a∈Γ

{a0, . . . , ala},⋃

(a,b)∈Γ 2

Hba) .

3 Refining Dynamics of Gene Regulatory Networks

We present two methods which aim at narrowing the set of dynamics of a ProcessHitting for a GRN: the first one is based on cooperativity between genes, theother one deals with the knowledge of the stable states.

3.1 Cooperative Hits

Given two genes c and f regulating a gene a, the action of c on a may dependon the level of f : there exists a cooperativity between c and f on a. In discreteframeworks, the cooperativity is often described by a boolean function betweengenes levels [1,19]. We show how to build cooperativity within Process Hitting.

Let (Σ,L,H) be a Process Hitting and σ ⊂ Σ be a set of sorts cooperatingon a given process ak to make it bounce to ak′ . The set of all states of thecooperating sorts is denoted by S =

z∈σ Lz. The subset of states where thecooperativity is effective is defined by ⊤ ⊂ S.

For applying this cooperation, a new sort is added to the Process Hitting.This sort is called a cooperative sort and is refered as υ. The set of processes ofsort υ is defined by Lυ = {υς , ∀ς ∈ S}. Each process zi of sort z ∈ σ hits processesυς of the cooperative sort υ where zi /∈ ς to make it bounce to υς[zi/zj ], zj ∈ ς.We denote Hσ the set of such actions (4). In this way, the process of sort υreflects the current state of its representatives.

8 L. Pauleve, M. Magnin and O. Roux

The cooperativity between υ processes is added into the Process Hitting byreplacing hits Hcoop from processes of sorts present in σ to ck (5) by hits H′

coop

from processes of the cooperative sort υ selected in ⊤ (6).

Hσ = {zi → υς � υς[zi/zj ] | ∀z ∈ σ, ∀(zi, zj) ∈ L2z, ∀υς ∈ Lυ, zj ∈ ς} (4)

Hcoop = {zi → ak � ak′ ∈ H | ∀z ∈ σ} (5)

H′coop = {υς → ak � ak′ | ∀ς ∈ ⊤} . (6)

The resulting Process Hitting is (Σ ∪{υ}, L×Lυ, (H\Hcoop)∪Hσ ∪H′coop).

Example 1. Let ({f, c, a}, {f0, f1} × {c0, c1} × {a0, a1},H) be a Process Hittingwhere {f1 → a0 � a1, c0 → a0 � a1} ⊂ H. The creation of a cooperativity be-tween f1 and c0 on a0 (σ = {f, c}, ⊤ = {f1c0}) is illustrated by Figure 2.

c0 c1

f1

f0

a0

a1

f1

f0

c0 c1

a0

a1

fc01 fc00 fc11 fc10

Fig. 2. Construction of a cooperative hit between f1 and c0 on a0 (thick lines):σ = {f, c}, ⊤ = {f1c0}, υ = fc. fc01 stands for the process corresponding to the statef0c1 of the cooperating processes σ.

3.2 Stable State Pattern

Given a Process Hitting (Σ,L,H), we prove the |Σ|-cliques of its Hitless Graphare exactly its stable states. Thus, stable states may be created by removingfrom the Process Hitting the very hits that make such patterns appear.

Theorem 1. Let PH = (Σ,L,H) be a Process Hitting and PH its HitlessGraph. A state s ∈ L is stable if and only if s is a |Σ|-clique for PH.

Proof. By definition, next(s) = ∅ if and only if there is no hit between anycouple of processes present in s. This is equivalent to have s a clique of PH.

Figure 3 shows an instance of Process Hitting having one stable state.We outline an algorithm for finding the n-cliques of a Hitless Graph PH =

(V,E) where n = |Σ|.Thanks to Prop. 1, we split V into n partitions corresponding to each process:

V = ∪a∈ΣVa, Va ⊆ La. If one of this partition is empty, there can not be n-cliques as it requires to have at least one vertex in each partition. We will assumeVa 6= ∅, ∀a ∈ Σ.

Refining Dynamics of Gene Regulatory Networks 9

For each partition a ∈ Σ and each vertex ai ∈ Va, we define Ebai

= {bj ∈Vb | (ai, bj) ∈ E} for each other partition b ∈ Σ, b 6= a, the set of vertices in Vb

related to ai. If there exists b ∈ Σ such that Ebai

= ∅, the vertex ai is removedfrom candidates as it can not belong to a n-clique. Finally, we set Ea

ai= {ai}.

Once this pruning is performed, we enumerate potential n-cliques. To reducethis enumeration, we choose the partition a sharing the least number of edges.For each vertex ai ∈ Va we test for all s ∈

b∈Σ Ebai

if s is a clique of PH.

For instance in Figure 3(b), a1 is removed from the Hitless Graph (Eba1

= ∅),the partition associated to c is chosen (involved into only 4 edges), two statesare tested: a0b0c1 and a2b1c0 and the latter reveals to be the only 3-clique.

a0 a1 a2

b0

b1

c0 c1

a0 a1 a2

b0

b1

c0 c1

(a) Hypergraph (b) Hitless Graph

Fig. 3. A Process Hitting represented by its Hypergraph (a) and its Hitless Graph (b).The Hitless Graph contains only one 3-clique between a2, b1 and c0 (thick lines): thisis the only stable state of this system.

4 From Process Hitting to Rene Thomas’ Parameters

A Rene Thomas’ discrete parameter gives the attractor levels for a gene whenits regulators are in a given configuration. Many frameworks and tools dedicatedto the study of GRNs take the full set of Rene Thomas’ parameters as essentialinput [1,9,11]. In this section, we give a formal method to infer Rene Thomas’parameters for a GRN modelled in the Process Hitting framework.

Let (Γ,E+, E−) be a GRN graph. A Rene Thomas’ parameter Ka,A,B , a ∈Γ,A∪B ⊆ Γ, A∩B = ∅, gives the interval of attracting levels for a when genesin A are activating a and genes in B are inhibiting a. In this configuration, ifthe level of a is in Ka,A,B , then it will never change; otherwise the level of a willtend to a level in Ka,A,B .

Let (Σ,L,H) be a Process Hitting where sorts are standing either for genesor for cooperative sorts, i.e. Σ = Γ ∪ {σ1, . . . , σu} with ∀1 ≤ v ≤ u, σv ⊂ Γ .Let Ka,A,B be the Rene Thomas’ parameter to infer. For each sorts b ∈ A ∪ B,its context Cb

a,A,B is defined as the subset of processes Lb imposed by the ReneThomas’ parameter: if b ∈ A (resp. B) only processes corresponding to positive(resp. negative) effective levels (Def. 2) are allowed. For each process b ∈ Γ

10 L. Pauleve, M. Magnin and O. Roux

not regulating a (i.e. b /∈ A ∪ B), its context Cba,A,B is simply Lb (7). The

context Cσa,A,B for cooperative sorts σ ∈ {σ1, . . . , σu} is the set of states of its

representatives in their context (8).

∀b ∈ Γ, Cba,A,B =

levels+(b, a) if b ∈ A,

levels−(b, a) if b ∈ B,

Lb otherwise.

(7)

∀σ ∈ {σ1, . . . , σu}, Cσa,A,B = {σς | ∀ς ∈

b∈σ

Cba,A,B} . (8)

We denote Ha,A,B the subset of the set of actions H on a that may beperformed by processes of the context of any sort (9). A process of sort a isreachable if it belongs to the context of a or is the result of any action in Ha,A,B .The set of such processes is noted L?

a,A,B (10). The set of reachable processesof sort a not hit by any other processes is noted L∗

a,A,B (11). Thus, as longas present processes are in the context of their sort, if a process of sort a isin L∗

a,A,B , it will not be bounced. In this way, L∗a,A,B is called the set of focal

processes of a.

Ha,A,B = {bi → aj � ak ∈ H | bi ∈ Cba,A,B ∧ aj ∈ Ca

a,A,B} (9)

L?a,A,B = Ca

a,A,B ∪ {ak | ∀bi → aj � ak ∈ Ha,A,B} (10)

L∗a,A,B = L?

a,A,B \ {aj | ∀bi → aj � ak ∈ Ha,A,B} . (11)

Finally, we check that the focal processes are attractors, i.e. all actionsHa,A,B

make processes of sort a bounce in the direction of the focal processes. If such acondition is satisfied, the focal processes correspond to the value of the requestedRene Thomas’ parameter. We point up that all these operations are linear withthe number of actions in the Process Hitting.

Condition 1 (Focal processes are attractors). ∀bi → aj � ak ∈ Ha,A,B , ∀af ∈L∗a,A,B , |f − k| < |f − j| .

Property 3. If L∗a,A,B satisfies Cond. 1, L∗

a,A,B is an interval.

Proof. If L∗a,A,B = {af , . . . , af ′} is not an interval, there exists bi → aj � ak ∈

Ha,A,B such that f < j < f ′. If Cond. 1 applies, we have |f − k| < |f − j| ⇒k < j ⇒ |f ′ − k| > |f ′ − j| which contradicts Cond. 1.

Theorem 2. If L∗a,A,B 6= ∅ and Cond. 1 holds, then Ka,A,B = L∗

a,A,B .

Proof. By construction of L∗a,A,B and application of Cond. 1 and Prop. 3, it

immediately appears that if L∗a,A,B 6= ∅, it is the set of attracting levels for a.

Consequently, there might exist configurations without any correspondencewith Rene Thomas’ parameters. First, L∗

a,A,B = ∅ means the gene a is unstablein the fixed context, i.e. its level is changing forever. Second, Cond. 1 is violatedwhen there exists opposite focal processes, i.e. the fate of a is not deterministic.

Refining Dynamics of Gene Regulatory Networks 11

One of the main reasons for non-determinism of Process Hitting is the absenceof cooperativity between hits to a same target which may then independentlybe bounced to both higher and lower processes. We leave as an open questionthe problem to know whether such unstable and/or non-deterministic dynamicsare biologically relevant.

5 Temporal and Stochastic Parameters

Further dynamics refinements may be achieved by taking into account the tem-poral and stochastic dimensions of biological reactions. On the one hand, we mayconsider the probability of a reaction to occur at a given state. By introducingstochastic parameters into discrete models, we aim at computing the probabil-ity of observing an expected behaviour. On the other hand, because they arefaster, some reactions always apply before others. By introducing temporal pa-rameters into discrete models, we aim at reducing their dynamics to match suchbehaviours.

5.1 From Process Hitting to the Stochastic π-Calculus

The Stochastic π-Calculus [20] adds the capability to attach use rates to channelsand internal actions of the π-Calculus. This gives a natural introduction fortemporal and stochastic aspects in our Process Hitting framework.

A use rate controls both the duration and the probability of a reaction (com-munication on a channel or internal action). It is associated to a probabilitydistribution for firing reaction along the time. The usual probability distribu-tion is the exponential one, allowing efficient simulations through a Gillespie-likealgorithm [12,21]. This is the one we consider for the rest of this paper.

The probability along time t of firing a reaction with use rate r is given byF (t) = 1− e−rt. The average duration of this reaction is r−1 with a variance ofr−2. When x reactions are possible having use rates of r1, . . . , rx respectively,the probability that the yth reaction is fired is given by

ryr1+···+rx

.The translation of Process Hitting (Σ,L,H) into the Stochastic π-Calculus

is the same as the one presented in Section 2.4. Additionally, to each channelγα, or internal action τα, a use rate rα is attached.

5.2 Stochasticity Absorption

Use rates are both temporal and stochastic parameters. Nonetheless, these twoaspects are closely tied: the lower a use rate is, the higher the variance around itsmean duration is. We introduce a stochasticity absorption factor to control thisvariance to favour either the stochastic or the temporal behaviour of an action.

We propose to replace the exponential distribution of a reaction with a rater by the distribution of the sum of sa random variables each having an ex-ponential distribution of parameter r.sa. The resulting probability distributionis also known as the Erlang distribution. The average duration is unchanged:

12 L. Pauleve, M. Magnin and O. Roux

(r.sa)−1sa = r−1, but the variance is divided by sa: (r.sa)−2sa = r−2sa−1. sastands for the stochasticity absorption factor. Based on the previously presentedtranslation from the Process Hitting to the Stochastic π-Calculus, we supply asimple method to achieve this stochasticity absorption factor which do not re-quire to adapt simulation algorithms based on the memoryless property of theexponential law [22].

Basically, to each channel γα, or internal action τα, a use rate rα and astochasticity absorption factor saα is attached. To each component α of the sumdefined by the π-Calculus process Ai in the expressions (2),(3), a counter cα isattached, initially, cα = 1. This counter is given as a parameter for Ai. As longas this counter is not equal to saα, Ai is restarted and the counter is incrementedby one. When the counter reaches the stochasticity absorption factor value, thenext process replaces Ai, having all its counters reset to 1. Let (Σ,L,H) be aProcess Hitting, for each process ai of PH, a π-Calculus process Ai is definedas follows.

Ai(c) ::=∑

α=ai → bj � bk∈H

a 6=b

!γα.Ai(c)

+∑

α=bj → ai � ak∈H

a 6=b

[cα < saα]?γα.Ai(c[cα + 1]) + [cα = saα]?γα.Ak(1)

+∑

α=ai → ai � ak∈H

[cα < saα]τα.Ai(c[cα + 1]) + [cα = saα]τα.Ak(1)

where c = c1, . . . , cn with n = |{bj → ai � ak ∈ H}|. c[cα + 1] = c1, . . . , cα +1, . . . , cn. Ak(1) is an abbreviation for the recursive call to Ak with all parametersset to 1. [cond]π.P stands for an action π enabled only when cond is satisfied.

6 Applications

6.1 Metazoan Segmentation

In this section, we illustrate our method and its benefits on a case study inwhich our aim is to control the final state of the corresponding GRN. The GRNwe chose has been established in silico by Francois et al. [23] but in a differ-ential equations framework. It aims at generalizing a common motif present inbiological segmentation networks such as the Drosophila.

The GRN (Figure 4(a)) is composed of three genes. A wavefront gene factivates the gap-gene a whose products are responsible or stripes. Gene f alsoactivates a gene c whose products repress the gene a. The auto-inhibition ofc generalizes a chain of repressors on a. The apparition of stripes has to beregular. We attach to each gene two processes representing their qualitative levels(missing or present) — for instance c0 (absence) and c1 (presence) are processes

Refining Dynamics of Gene Regulatory Networks 13

for c. When f switches off, c goes to process c0 and a has two fates, ending eitherat process a0 or a1. We are interested in controlling the final process for a.

The Process Hitting for generalised dynamics of the GRN (Section 2) iscomputed first. Figure 4(b) shows its hypergraph. The specification of dynamicsimplies two cooperative hits in the Process Hitting: first, c0 needs productsof f to bounce to process c1; second, expression of a only increases if bothf activates it (i.e. process f1 is present) and c does not inhibit it (i.e. c0 ispresent). Consequently, we create a cooperative sort fc reflecting the state of fand c (Section 3.1) and replace the independent hits from c0 and f1 to c0 and a0by hits from fc10. The resulting Process Hitting is represented in Figure 5(a).

By looking at the Hitless Graph of the Process Hitting (Figure 5(b)), onlyone stable state is present: f0c0fc00a0. The stability of the state f0c0fc00a1 iscontrolled by the absence of inhibition by f0 on a1. By removing the actionf0 → a1 � a0 from the Process Hitting, we make the state f0c0fc00a1 stable. Thefull set of corresponding Rene Thomas’ parameters for the genes a and c isinferred by applying the method depicted in Section 4. We get:

Ka,∅,{a,c,f} = 0 Ka,{a,c},{f} = 1 Kc,∅,{c,f} = 0Ka,{a},{c,f} = 0 Ka,{a,f},{c} = 0 Kc,{c},{f} = 0Ka,{c},{a,f} = 0 Ka,{c,f},{a} = 1 Kc,{f},{c} = 0Ka,{f},{a,c} = 0 Ka,{a,c,f},∅ = 1 Kc,{c,f},∅ = 1 .

We are interested in controlling the final process of sort a — either a0 ora1 — when f goes down to f0. Looking at the Process Hitting hypergraph onFigure 5(a) and considering f0 is present, we deduce that the more c1 is present,the more a1 may be hit by c1 to bounce to a0; similarly, the more fc10 is present,the more a0 may be hit by it to bounce to a1. We tune actions only triggered byf0: we reduce the presence of c1 by increasing the rate of the action f0 → c1 � c0and extend the presence of fc10 by reducing the rate of f0 → fc10 � fc00. Thisleads to an increase of the probability for a to bounce to process a1.

Finally, to obtain regular stripes, we set a high stochasticity absorption factorto actions responsible of the bounces of processes of sort c and a when f1 ispresent. Figure 6 plots the evolution of the genes a,c and f during a simulationunder SPiM [24] of the Process Hitting illustrated by Figure 5(a) with initialstate f1c0fc10a0 and a fast rate for the action f0 → c1 � c0 compared to the rateof c1 → a1 � a0. The rate values have been arbitrarily choosen and respect theinfered relations between them. Appendix B.1 details the Process Hitting usedfor the simulation.

From the obtained simulation trace, we observe that f0 hits c1 before c1 hadtime to hit a1: the final state is then f0c0fc00a1.

Thanks to the Process Hitting framework, it has been easy to build a qual-itative model of the biological system by refining the generalised dynamics ofthe GRN. Using a simple reasoning on the Process Hitting structure, relationbetween regulation delays to favour a final state have been infered. These resultsare coherent with those obtained using differential equations as done in [23].

14 L. Pauleve, M. Magnin and O. Roux

f

c ac1

c0

a1

a0

f0

f1

(a) (b)

Fig. 4. The starting Gene Regulatory Network graph (a), arrow-ended edges representthe positive regulations, and bar-ended edges the negative ones. All regulation thresh-olds are 1. The Process Hitting (b) for its generalized dynamics. Cooperativity betweenf1 and c0 on a0 and c0 will be applied in the same way as in example. 1.

c1

c0

f0

f1

a1

a0

fc11

fc10

fc01

fc00

c0

f0

a1

a0

fc11 fc10 fc01 fc00

(b)(a)

Fig. 5. The final Process Hitting (a) resulting from the refinement of the generalizeddynamics depicted on Fig. 4(b). Cooperativity between f1 and c0 on a0 and c0 hasbeen built in the same way as in example 1. Absence of hit from f0 to a1 (dashed lines)controls the presence of the relation between f0 and a1 in the Hitless Graph (b). If sucha relation exists, two 4-cliques are presents: c0f0fc00a0 and c0f0fc00a1 (thick lines).

Refining Dynamics of Gene Regulatory Networks 15

0

1

0 5 10 15 20 25 30 35 40

0

1

0 5 10 15 20 25 30 35 40

0

1

0 5 10 15 20 25 30 35 40

Fig. 6. Simulation of the Process Hitting for segmentation: evolution of the expressionsof the gap-gene a (top), the autonomous clock c (middle) and the wavefront f .

6.2 ERBB Receptor-Regulated G1/S Transition

The aim of this section is to demonstrate the scalability of the refining approachon Process Hittings modelling large GRNs.

The selected GRN relates regulations between 20 genes. This GRN modelsthe ERBB receptor-regulated G1/S transition involved in the breast cancer. Ithas been extracted from published data by Sahin et al. [25] and is reproducedin Figure 7. This network acts as an activation cascade for the gene pRB whichcontrols the G1/S transition involved in cell divisions. The gene EGF then actsas an input of this cascade: when expressed, pRB will be potentially activated.Based on the literature, Sahin et al. have also established a set of logical rulescontrolling the activation of the various genes present in the network.

Starting from the GRN, its generalised dynamics expressed in Process Hittingis first computed. Then, cooperations between the different sorts are built fromthe logical rules. The Process Hitting obtained contains 670 actions and standsfor 264 (≈ 2.1019) states that has hopefully not be built. This model is fullydetailled in Appendix B.2.

The computation of all the stable states present in the dynamics is doneusing the algorithm sketched by Section 3.2. It results in 5 stables states (alsodetailled in the appendix) that are computed in less than one second.

It is worth noticing that no assumption is made on the initial state of thesystem. All the stable states of the model are computed. This is a major differ-ence with the approach presented in [25] where only dynamics starting from afixed state can be studied.

7 Conclusion

We introduced the Process Hitting framework for modelling qualitative dynamicsof GRNs with temporal and stochastic features. Temporal and stochastic param-

16 L. Pauleve, M. Magnin and O. Roux

EGF

ERBB2ERBB1 ERBB3

ERBB1 2 ERBB1 3 ERBB2 3 IGF1R

MEK1AKT1

ERalpha MYC

CycD1 p27 p21 CycE1

CDK6 CDK2CDK4

pRB

Fig. 7. ERBB receptor-regulated G1/S transition GRN reproduced from

Refining Dynamics of Gene Regulatory Networks 17

eters determine probabilities, durations and temporal variance of reactions in themodel. We exhibited a direct translation from Process Hitting to the Stochasticπ-Calculus. Detection of stable states and inference of Rene Thomas’ parame-ters for dynamics derive from this framework. The methods we offered work bysuccessive refinements of generalised dynamics for GRNs, by specifying both thecooperativity between genes and the expected stable states. We illustrated thismethod by inferring temporal parameters for the dynamics of a GRN generaliz-ing metazoan segmentation processes (with the aim of controlling its final state).The scalability of the presented approach has been experimented on a ProcessHitting modelling a GRN composed of 20 genes and computing its stable states.

The Process Hitting brings a formal framework for progressively addingknowledge of the dynamics of a GRN by refining an abstracted behaviour.The compositionality of the framework and the presence of particular structurepatterns lead to scalable methods for dynamics analyses (stable states, ReneThomas’ parameters, etc.). Mainly, thanks to these Process Hitting patterns, itbecomes possible to perform a local analysis, which has the major advantage toprevent us from exploring the full state and parameter space.

In future works, we aim at identifying more Process Hitting patterns leadingto the emergence of particular behaviours (e.g. oscillations) and especially hybridpatterns coupling both discrete structure and continuous temporal and stochas-tic parameters. The verification of Process Hittings could be performed by usinga translation into Petri Nets or into PRISM. Translating Process Hittings intomore sophisticated process aglebras (Beta Workbench, Bio-PEPA, etc.) may alsobe of interest. Finally, techniques have to be developed to infer automaticallytemporal and stochastic parameters of Process Hittings modelling GRNs.

Supplementary Material

The Process Hitting compiler to SPiM, a stable states computer and presentedmodels are available at the following URL:http://www.irccyn.ec-nantes.fr/~pauleve/processhitting-refining.tar.

gz .

References

1. Richard, A., Comet, J.P., Bernot, G.: Formal Methods for Modeling BiologicalRegulatory Networks. In: Modern Formal Methods and Applications. (2006) 83–122

2. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-Based Modellingof Cellular Signalling. In: CONCUR 2007 Concurrency Theory. (2007) 17–41

3. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochasticname-passing calculus to representation and simulation of molecular processes.Inf. Process. Lett. 80(1) (2001) 25–31

4. Kuttler, C., Niehren, J.: Gene regulation in the pi calculus: Simulating cooper-ativity at the lambda switch. Transactions on Computational Systems Biology4230(VII) (2006) 24–55

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5. Blossey, R., Cardelli, L., Phillips, A.: A compositional approach to the stochasticdynamics of gene networks. Transactions in Computational Systems Biology 3939

(Jan 2006) 99–1226. Popova-Zeugmann, L., Heiner, M., Koch, I.: Time petri nets for modelling and

analysis of biochemical networks. Fundamenta Informaticae 67(1) (2005) 149–1627. Heiner, M., Gilbert, D., Donaldson, R.: Petri Nets for Systems and Synthetic

Biology. In: Formal Methods for Computational Systems Biology. (2008) 215–2648. Rizk, A., Batt, G., Fages, F., Soliman, S.: On a continuous degree of satisfaction

of temporal logic formulae with applications to systems biology. In Heiner, M.,Uhrmacher, A., eds.: Computational Methods in Systems Biology. Volume 5307 ofLecture Notes in Computer Science. Springer Berlin / Heidelberg (2008) 251–268

9. Siebert, H., Bockmayr, A.: Incorporating time delays into the logical analysis ofgene regulatory networks. In Priami, C., ed.: Computational Methods in SystemsBiology. Volume 4210 of Lecture Notes in Computer Science. Springer Berlin /Heidelberg (2006) 169–183

10. Alur, R., Belta, C., Kumar, V., Mintz, M., Pappas, G.J., Rubin, H., Schug, J.:Modeling and analyzing biomolecular networks. Computing in Science and Engi-neering 4(1) (2002) 20–31

11. Ahmad, J., Bernot, G., Comet, J.P., Lime, D., Roux, O.: Hybrid modelling anddynamical analysis of gene regulatory networks with delays. Complexus 3(4) (2006)231–251

12. Phillips, A., Cardelli, L.: Efficient, correct simulation of biological processes in thestochastic pi-calculus. In: Computational Methods in Systems Biology. Volume4695 of LNCS., Springer (2007)

13. Dematte, L., Priami, C., Romanel, A.: The Beta Workbench: a computational toolto study the dynamics of biological systems. Brief Bioinform (2008) bbn023

14. Ciocchetta, F., Hillston, J.: Bio-pepa: A framework for the modelling and analysisof biological systems. Theoretical Computer Science 410(33-34) (2009) 3065 –3084

15. Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for auto-matic verification of probabilistic systems. In: 12th International Conference onTools and Algorithms for the Construction and Analysis of Systems. Volume 3920of LNCS., Springer (2006)

16. Norman, G., Palamidessi, C., Parker, D., Wu, P.: Model checking probabilisticand stochastic extensions of the π-calculus. IEEE Trans. on Software Engineering35(2) (2009) 209–223

17. Bernot, G., Cassez, F., Comet, J.P., Delaplace, F., Muller, C., Roux, O.: Seman-tics of biological regulatory networks. Electronic Notes in Theoretical ComputerScience 180(3) (2007) 3 – 14

18. Milner, R.: A calculus of mobile processes, parts. I and II. Information andComputation 100 (1992) 1–77

19. Bernot, G., Comet, J.P., Khalis, Z.: Gene regulatory networks with multiplexes. In:European Simulation and Modelling Conference Proceedings. (Oct 2008) 423–432

20. Priami, C.: Stochastic π-Calculus. The Computer Journal 38(7) (1995) 578–58921. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The

Journal of Physical Chemistry 81(25) (1977) 2340–236122. Priami, C.: Stochastic π-calculus with general distributions. In: Proc. of the 4th

Workshop on Process Algebras and Performance Modelling, CLUT. (1996) 41–5723. Francois, P., Hakim, V., Siggia, E.D.: Deriving structure from evolution: metazoan

segmentation. Mol Syst Biol 3 (2007)

Refining Dynamics of Gene Regulatory Networks 19

24. Phillips, A.: SPiM - Stochastic Pi Machine. (SPiM) http://research.microsoft.com/en-us/projects/spim.

25. Sahin, O., Frohlich, H., Lobke, C., Korf, U., Burmester, S., Majety, M., Mattern,J., Schupp, I., Chaouiya, C., Thieffry, D., Poustka, A., Wiemann, S., Beissbarth,T., Arlt, D.: Modeling ERBB receptor-regulated G1/S transition to find noveltargets for de novo trastuzumab resistance. BMC Systems Biology 3(1) (2009)

A Process Hitting Related Tools

This appendix briefly presents currently available implementations of tools ma-nipulating Process Hittings. They are available at the following URL:http://www.irccyn.ec-nantes.fr/~pauleve/processhitting-refining.tar.

gz .

Implemented in the OCAML language, these tools have a command-line userinterline.

A.1 Process Hitting Specification

A basic language has been setup to specify Process Hitting using a text file. Mainfeatures are presented here, more details can be found in the provided archive.Full examples of Process Hitting specifications are given in Appendix B.

Sort definition A sort is declared by giving the process with the highest rank(i.e. ala for the sort a, with the notations used in Section 2).

process a X

Action specification An action ai → bj � bk is added by the following instruction:

a i -> b j k @ rate ~ absorption

Generalised dynamics of GRNs The GRN macro computes the generalised dy-namics of the specified GRN according to Section 2.5. An activating (resp. in-

hibiting) edge aX−→ b from gene a to gene b active with a threshold X is noted

as a X -> + b (resp. a X -> - b).

Hereafter is an instance of the use of the GRN macro for the GRN having

activating edges a1−→ b and a

2−→ a, and inhibiting edge b

1−→ b.

GRN([a 1 -> + b; b 1 -> - a; a 2 -> + a ])

Refinement: cooperations The COOPERATIVITY([a1;...;an] -> b j k, [s1;...;sp])

macro creates a cooperative sort σ = {a1, . . . , an} for the bounce bj � bk. Thiscooperation is effective for every state s1, . . . , sp.

The following instruction creates a cooperativity between sorts a and b tobounce process c0 to c1 only if a1b0 or a0b1 are present.

COOPERATIVITY([a;b] -> c 0 1, [[1;0];[0;1]])

20 L. Pauleve, M. Magnin and O. Roux

Refinement: action removing An action ai → bj � bk can be deleted by using themacro RM:

RM({a i -> b j k})

A.2 Compiler to SPiM

This tool translates a Process Hitting specification into a SPiM model accordingto Section 5.2.

phc -spim <model.ph> <output.spi>

A.3 Stable States Listing

The stable states of a Process Hitting are determined used an implementationof the algorithm sketched in Section 3.2. This algorithm computes the n-cliquesof the hitless graph for the given Process Hitting, where n is the number ofsorts. The efficiency of this computation heavily relies on the order of sortswhen building cliques. Currently, basic heuristics are used to select the order ofthe sorts. More sophisticated analyses may conduct to dramatically improve theefficiency of this algorithm.

This tool is compiled into the executable ph-stable-states and takes asargument the filename of the Process Hitting specification :

ph-stable-states <model.ph>

B Process Hitting Examples

This appendix details the Process Hittings used in Section 6. They are specifiedin the language presented in the previous appendix.

B.1 Metazoan Segmentation

The following Process Hitting models the metazoan segmentation presentedin Section. 6.1. Actions are specified separately and rates have been assignedto values matching the relations infered in the application case study. Thedirective sample and initial state instructions are of use for SPiM only. A result ofthe execution of this Process Hitting translated into SPiM is given by Figure 6.

directive sample 40.

process a 1 process c 1 process f 1process fc 3 (* cooperative sort {f,c} *)c 1 -> fc 0 1 @5.c 1 -> fc 2 3 @5.c 0 -> fc 1 0 @10.c 0 -> fc 3 2 @5.f 1 -> fc 0 2 @10.f 1 -> fc 1 3 @10.f 0 -> fc 2 0 @0.1f 0 -> fc 3 1 @0.1

Refining Dynamics of Gene Regulatory Networks 21

(* actions on c *)fc 2 -> c 0 1 @0 .5~50 (* only if (f1 ,c0) *)c 1 -> c 1 0 @0 .5~50(* actions on a *)fc 2 -> a 0 1 @1.~50 (* only if (f1 ,c0) *)c 1 -> a 1 0 @1.~50(* actions on f *)f 1 -> f 1 0 @0 .034~100 (* auto -off *)f 0 -> c 1 0 @0.1

init ial state f 1, c 0, a 0

B.2 ERBB Receptor-Regulated G1/S Transition

The following Process Hitting results from the case study presented in Sec-tion 6.2. It starts by specifying the GRN depicted by Figure 7 to compute itsgeneralised dynamics. The logical rules setup by Sahin et al. [25] are then appliedby using sorts cooperativity.

This Process Hitting contains 670 actions and 264 (≈ 2.1019) states. Only5 stable states exist and are determined in less than a second using the toolpresented in the previous appendix.

Below is the list of stable states present in the Process Hitting. For eachstable state, only genes at level 1 are written.

– AKT1, CDK2, CDK4, CDK6, CycD1, CycE1, EGF, ERBB1, ERBB1 2, ERBB1 3, ERBB2,

ERBB2 3, ERBB3, ERalpha, MEK1, MYC, pRB.

– AKT1, CDK2, CDK4, CDK6, CycD1, CycE1, ERalpha, IGF1R, MEK1, MYC, pRB.

– AKT1, CDK2, CycE1, EGF, ERBB1, ERBB1 2, ERBB1 3, ERBB2, ERBB2 3, ERBB3, ERal-

pha, MEK1, MYC.

– AKT1, CDK2, CycE1, ERalpha, IGF1R, MEK1, MYC.

– ∅ (all genes have level 0).

process AKT1 1process CDK2 1 process CDK4 1 process CDK6 1process CycD1 1 process CycE1 1process EGF 1 process ERalpha 1process ERBB1 1 process ERBB1_2 1 process ERBB1_3 1process ERBB2 1 process ERBB2_3 1 process ERBB3 1process IGF1R 1 process MEK1 1 process MYC 1process p21 1 process p27 1process pRB 1

GRN([ERBB2_3 1 -> + AKT1; ERBB2_3 1 -> + MEK1; ERBB2_3 1 -> - IGF1R;ERBB2 1 -> + ERBB2_3; ERBB2 1 -> + ERBB1_2; ERBB3 1 -> + ERBB2_3;ERBB3 1 -> + ERBB1_3;CycE1 1 -> + CDK2;MEK1 1 -> + CycD1; MEK1 1 -> + ERalpha; MEK1 1 -> + MYC;CDK4 1 -> + pRB; CDK4 1 -> - p21; CDK4 1 -> - p27;ERalpha 1 -> + CycD1; ERalpha 1 -> + IGF1R; ERalpha 1 -> + p21;ERalpha 1 -> + MYC; ERalpha 1 -> + p27;MYC 1 -> + CycE1; MYC 1 -> - p21; MYC 1 -> + CycD1; MYC 1 -> - p27;CDK6 1 -> + pRB;ERBB1 1 -> + ERBB1_2; ERBB1 1 -> + ERBB1_3; ERBB1 1 -> + AKT1;ERBB1 1 -> + MEK1;IGF1R 1 -> + AKT1; IGF1R 1 -> + MEK1;

22 L. Pauleve, M. Magnin and O. Roux

ERBB1_3 1 -> + AKT1; ERBB1_3 1 -> + MEK1;p27 1 -> - CDK2; p27 1 -> - CDK4;CDK2 1 -> - p27; CDK2 1 -> + pRB;p21 1 -> - CDK2; p21 1 -> - CDK4;CycD1 1 -> + CDK4; CycD1 1 -> + CDK6;EGF 1 -> + ERBB1; EGF 1 -> + ERBB2; EGF 1 -> + ERBB3;AKT1 1 -> + CycD1; AKT1 1 -> + MYC; AKT1 1 -> - p27; AKT1 1 -> + ERalpha;AKT1 1 -> + IGF1R; AKT1 1 -> - p21;ERBB1_2 1 -> + AKT1; ERBB1_2 1 -> + MEK1;

])

COOPERATIVITY([ ERBB1;ERBB2] -> ERBB1_2 0 1, [[1;1]])COOPERATIVITY([ ERBB1;ERBB3] -> ERBB1_3 0 1, [[1;1]])COOPERATIVITY([ ERBB2;ERBB3] -> ERBB2_3 0 1, [[1;1]])

COOPERATIVITY([ ERBB2_3;AKT1] -> IGF1R 0 1, [[0;1]])COOPERATIVITY([ ERBB2_3;ERalpha] -> IGF1R 0 1, [[0;1]])COOPERATIVITY([AKT1;ERalpha] -> IGF1R 1 0, [[0;0]])

COOPERATIVITY([AKT1;MEK1] -> ERalpha 1 0, [[0;0]])COOPERATIVITY([AKT1;MEK1;ERalpha] -> MYC 1 0, [[0;0;0]])COOPERATIVITY([ ERBB1;ERBB1_2;ERBB1_3;ERBB2_3;IGF1R] -> AKT1 1 0,

[[0;0;0;0;0]])COOPERATIVITY([ ERBB1;ERBB1_2;ERBB1_3;ERBB2_3;IGF1R] -> MEK1 1 0,

[[0;0;0;0;0]])COOPERATIVITY([ CycE1;p21;p27] -> CDK2 0 1, [[1;0;0]])COOPERATIVITY([ CycD1;p21;p27] -> CDK4 0 1, [[1;0;0]])COOPERATIVITY([ ERalpha;MYC;AKT1;MEK1] -> CycD1 0 1, [[1;1;1;0];[1;1;0;1]])COOPERATIVITY([AKT1;MEK1] -> CycD1 1 0, [[0;0]])COOPERATIVITY([ ERalpha;AKT1;MYC;CDK4] -> p21 0 1, [[1;0;0;0]])COOPERATIVITY([ ERalpha;CDK4;CDK2;AKT1;MYC] -> p27 0 1, [[1;0;0;0;0]])COOPERATIVITY([CDK2;CDK4;CDK6] -> pRB 0 1, [[0;1;1];[1;1;1]])RM({CDK2 0 -> pRB 1 0})RM({EGF 1 -> EGF 1 0}) (* prevent self -degradation (input) *)